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authorGravatar Tim Hosgood <timhosgood@gmail.com> 2021-07-29 00:37:33 +0100
committerGravatar Tim Hosgood <timhosgood@gmail.com> 2021-07-29 00:37:33 +0100
commit21c333e1af020733e4d7af5a636852c5ec0c6f45 (patch)
tree5034442341a31344ba08d879ab21992c193189c2
parentb490ab8d4675d6b25f22dde6abf63ccc4b4be8f4 (diff)
downloadega-21c333e1af020733e4d7af5a636852c5ec0c6f45.tar.gz
ega-21c333e1af020733e4d7af5a636852c5ec0c6f45.zip
finished II.2!
-rw-r--r--README.md2
-rw-r--r--ega2/ega2-2.tex73
2 files changed, 71 insertions, 4 deletions
diff --git a/README.md b/README.md
index a827d61..94c6f1a 100644
--- a/README.md
+++ b/README.md
@@ -70,7 +70,7 @@ Here is the current status of the translation, along with who is currently worki
- **Elementary global study of some classes of morphisms (EGA II)** ![EGAIIstatus](https://img.shields.io/badge/-130%2F205-yellow)
+ [x] 0. Summary _(@ryankeleti / proofread by @thosgood)_
+ [x] 1. Affine morphisms _(@ryankeleti)_
- + [ ] 2. Homogeneous prime spectra (~30 pages) _(@thosgood)_
+ + [x] 2. Homogeneous prime spectra _(@thosgood)_
+ [ ] 3. Homogeneous prime spectrum of a sheaf of graded algebras (~20 pages)
+ [x] 4. Projective bundles; Ample sheaves _(@thosgood)_
+ [x] 5. Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms _(@thosgood)_
diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
index 1ce0986..102c75a 100644
--- a/ega2/ega2-2.tex
+++ b/ega2/ega2-2.tex
@@ -1512,7 +1512,7 @@ for all $n\in\bb{Z}$.
\begin{env}[2.8.9]
\label{II.2.8.9}
Let $A$ and $A'$ be rings, and $\psi:A'\to A$ a ring homomorphism, defining a morphism $\Psi:\Spec(A)\to\Spec(A')$.
-Let $S'$ be a positively-graded $A'$-algebra, and set $S=S'\otimes_{A'}A$, which is evidently an $A$-algebra graded by the $S'_n\otimes_{A'}A$;
+Let $S'$ be a positively graded $A'$-algebra, and set $S=S'\otimes_{A'}A$, which is evidently an $A$-algebra graded by the $S'_n\otimes_{A'}A$;
the map $\varphi:s'\to s'\otimes1$ is then a graded ring homomorphism that makes the diagram \sref{II.2.8.5.1} commute.
Since $S_+$ is here the $A$-module generated by $\varphi(S'_+)$, we have $G(\varphi)=\Proj(S)=X$;
whence, setting $X'=\Proj(S')$, we have the commutative diagram
@@ -1681,10 +1681,77 @@ With the notation and hypotheses of \sref{II.2.8.1}, it follows from \sref{II.2.
\end{env}
-\subsection{Closed subschemes of a scheme $\Proj(S)$}
+\subsection{Closed subschemes of a scheme $\operatorname{Proj}(S)$}
\label{subsection:II.2.9}
\begin{env}[2.9.1]
\label{II.2.9.1}
-If
+If $\varphi:S\to S'$ is a homomorphism of graded rings, then we say that $\varphi$ is (TN)-\emph{surjective} (resp. (TN)-\emph{injective}, (TN)-\emph{bijective}) if there exists an integer $n$ such that, for $k\geq n$, $\varphi_k:S_k\to S'_k$ is \emph{surjective} (resp. \emph{injective}, \emph{bijective}).
+Instead of saying that $\varphi$ is (TN)-bijective, we sometimes say that it is a (TN)-\emph{isomorphism}.
\end{env}
+
+\begin{proposition}[2.9.2]
+\label{II.2.9.2}
+Let $S$ be a positively graded ring, and let $X=\Proj(S)$.
+\begin{enumerate}
+ \item[\rm{(i)}] If $\varphi:S\to S'$ is a (TN)-surjective homomorphism of graded rings, then the corresponding morphism $\Phi$ \sref{II.2.8.1} is defined on the whole of $\Proj(S')$, and is a closed immersion of $\Proj(S')$ into $X$.
+ If $\mathfrak{J}$ is the kernel of $\varphi$, then the closed subscheme of $X$ associated to $\Phi$ is defined by the quasi-coherent sheaf of ideals $\widetilde{\mathfrak{J}}$ of $\sh{O}_X$.
+ \item[\rm{(ii)}] Suppose further that the ideal $S_+$ is generated by a finite number of homogeneous elements of degree~$1$.
+ Let $X'$ be a closed subscheme of $X$ defined by a quasi-coherent sheaf of ideals $\sh{J}$ of $\sh{O}_X$.
+ Let $\mathfrak{J}$ be the graded ideal of $S$ given by the inverse image of $\Gamma_\bullet(\sh{J})$ under the canonical homomorphism $\alpha:S\to\Gamma_\bullet(\sh{O}_X)$ \sref{II.2.6.2}, and set $S'=S/\mathfrak{J}$.
+ Then $X'$ is the subscheme associated to the closed immersion $\Proj(S')\to X$ corresponding to the canonical homomorphism of graded rings $S\to S'$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+\begin{enumerate}
+ \item[\rm{(i)}] We can suppose that $\varphi$ is surjective \sref{II.2.9.1}.
+ Since, by hypothesis, $\varphi(S_+)$ generates $S'_+$, we have $G(\varphi)=\Proj(S')$.
+ Now, the second claim can be checked locally on $X$;
+ so let $f$ be a homogeneous element of $S_+$, and set $f'=\varphi(f)$.
+ Since $\varphi$ is a surjective homomorphism of graded rings, we immediately see that $\varphi_{(f')}:S_{(f)}\to S'_{(f')}$ is surjective, and that its kernel is $\mathfrak{J}_{(f)}$, which proves (i) \sref[I]{I.4.2.3}.
+ \item[\rm{(ii)}] By (i), we are led to proving that the homomorphism $\widetilde{j}:\widetilde{\mathfrak{J}}\to\sh{O}_X$ induced by the canonical injection $j:\mathfrak{J}\to S$ is an isomorphism from $\widetilde{\mathfrak{J}}$ to $\sh{J}$, which follows from \sref{II.2.7.11}.
+\end{enumerate}
+\end{proof}
+
+We note that $\mathfrak{J}$ is the \emph{largest} of the graded ideals $\mathfrak{J}'$ of $S$ such that $\widetilde{j}(\widetilde{\mathfrak{J'}})=\sh{J}$, since we can immediately show, using the definitions \sref{II.2.6.2}, that this equation implies that $\alpha(\mathfrak{J}')\subset\Gamma_\bullet(\sh{J})$.
+
+\begin{corollary}[2.9.3]
+\label{II.2.9.3}
+Suppose that the hypotheses of \sref{II.2.9.2}[(i)] are satisfied, and further that the ideal $S_+$ is generated by $S_1$;
+then $\Phi^*((S(n))\supertilde)$ is canonically isomorphic to $(S'(n))\supertilde$ for all $n\in\bb{Z}$, and so $\Phi^*(\sh{F}(n))$ is canonically isomorphic to $\Phi^*(\sh{F})(n)$ for every $\sh{O}_X$-module $\sh{F}$.
+\end{corollary}
+
+\begin{proof}
+This is a particular case of \sref{II.2.8.8}, taking \sref{II.2.5.10.2} into account.
+\end{proof}
+
+\begin{corollary}[2.9.4]
+\label{II.2.9.4}
+Suppose that the hypotheses of \sref{II.2.9.2}[(ii)] are satisfied.
+For the closed sub-prescheme $X'$ of $X$ to be integral, it is necessary and sufficient for the graded ideal $\mathfrak{J}$ to be prime in $S$.
+\end{corollary}
+
+\oldpage[II]{49}
+
+\begin{proof}
+Since $X'$ is isomorphic to $\Proj(S/\mathfrak{J})$, the condition is sufficient by \sref{II.2.4.4}.
+To see that it is necessary, consider the exact sequence $0\to\sh{J}\to\sh{O}_X\to\sh{O}_X/\sh{J}$, which gives the exact sequence
+\[
+ 0 \to \Gamma_\bullet(\sh{J}) \to \Gamma_\bullet(\sh{O}_X) \to \Gamma_\bullet(\sh{O}_X/\sh{J}).
+\]
+
+It suffices to prove that, if $f\in S_m$ and $g\in S_n$ are such that the image in $\Gamma_\bullet(\sh{O}_X/\sh{J})$ of $\alpha_{n+m}(fg)$ is zero, then the image of either $\alpha_m(f)$ or $\alpha_n(g)$ is zero.
+But, by definition, these images are sections of invertible $(\sh{O}_X/\sh{J})$-modules $\sh{L}=(\sh{O}_X/\sh{J})(m)$ and $\sh{L}'=(\sh{O}_X/\sh{J})(n)$ over the integral scheme $X'$;
+the hypothesis implies that the product of these two sections is zero in $\sh{L}\otimes\sh{L}'$ (\sref{II.2.9.3} and \sref{II.2.5.14.1}), and so one of them is zero by \sref[I]{I.7.4.4}.
+\end{proof}
+
+\begin{corollary}[2.9.5]
+\label{II.2.9.5}
+Let $A$ be a ring, $M$ an $A$-module, $S$ a graded $A$-algebra generated by the set $S_1$ of homogeneous elements of degree~$1$, $u:M\to S_1$ a surjective homomorphism of $A$-modules, and $\overline{u}:\bb{S}(M)\to S$ the homomorphism (of $A$-algebras) from the symmetric algebra $\bb{S}(M)$ of $M$ to $S$ that extends $u$.
+Then the morphism corresponding to $\overline{u}$ is a closed immersion of $\Proj(S)$ into $\Proj(\bb{S}(M))$.
+\end{corollary}
+
+\begin{proof}
+Indeed, $\overline{u}$ is surjective by hypothesis, and so it suffices to apply \sref{II.2.9.2}
+\end{proof}